Constructing Inverse Trigonometric Functions - Precalculus

1. What is the standard restricted domain for defining $\arccos(x)$ as an inverse of $\cos(x)$?

   $\left[0,\pi\right]$. On this interval, cosine is strictly decreasing from 1 to -1.

2. Identify whether $\tan(x)$ is increasing or decreasing on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.

   Increasing on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Derivative $\sec^2(x)>0$ on this interval confirms monotonic increase.

3. What is the relationship between the graphs of $y=f(x)$ and $y=f^{-1}(x)$?

   They are reflections across the line $y=x$. Inverse functions are symmetric about the line $y=x$.

4. Identify the principal-value identity for $\sin(\arcsin(x))$.

   $\sin(\arcsin(x))=x$ for $x\in\left[-1,1\right]$. Applying sine to its inverse returns the original value.

5. What identity expresses the inverse relationship on the domain of $\arccos$?

   $\cos(\arccos(x))=x$ for $x\in\left[-1,1\right]$. Composing a function with its inverse yields the identity function.

6. What is the simplified value of $\arccos\left(\cos\left(\frac{5\pi}{3}\right)\right)$?

   $\frac{\pi}{3}$. $\cos(\frac{5\pi}{3})=\frac{1}{2}$, and $\arccos(\frac{1}{2})=\frac{\pi}{3}$ in range.

7. Identify the restricted interval where $\tan(x)$ is strictly increasing and invertible.

   $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Tangent increases continuously without vertical asymptotes here.

8. What does it mean for a function to be one-to-one in terms of the horizontal line test?

   Every horizontal line intersects the graph at most once. This test verifies that each y-value corresponds to at most one x-value.

9. What restricted domain is used to define $\arccos(x)$ as the inverse of $\cos(x)$?

   $\left[0,\pi\right]$. On this interval, cosine decreases from 1 to -1, making it one-to-one.

10. Identify whether $\sin(x)$ is increasing or decreasing on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

    Increasing on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Derivative $\cos(x)>0$ on this interval confirms monotonic increase.

11. Identify the restricted interval where $\cos(x)$ is strictly decreasing and invertible.

    $\left[0,\pi\right]$. Cosine decreases monotonically from 1 to -1 on this interval.

12. What is the simplified value of $\arctan\left(\tan\left(\frac{3\pi}{4}\right)\right)$?

    $-\frac{\pi}{4}$. $\tan(\frac{3\pi}{4})=-1$, and $\arctan(-1)=-\frac{\pi}{4}$ in range.

13. Identify whether $\cos(x)$ is increasing or decreasing on $\left[0,\pi\right]$.

    Decreasing on $\left[0,\pi\right]$. Derivative $-\sin(x)<0$ on this interval confirms monotonic decrease.

14. Identify the principal-value identity for $\cos(\arccos(x))$.

    $\cos(\arccos(x))=x$ for $x\in\left[-1,1\right]$. Applying cosine to its inverse returns the original value.

15. What is the range of $\arctan(x)$?

    $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. This matches the restricted domain of tangent used to define its inverse.

16. What is the range of $\arccos(x)$?

    $\left[0,\pi\right]$. This matches the restricted domain of cosine used to define its inverse.

17. What identity expresses the inverse relationship on the domain of $\arctan$?

    $\tan(\arctan(x))=x$ for $x\in\left(-\infty,\infty\right)$. Composing a function with its inverse yields the identity function.

18. What is the domain of $\arcsin(x)$?

    $\left[-1,1\right]$. Since sine's range is $[-1,1]$, this becomes arcsine's domain.

19. What is the standard restricted domain for defining $\arcsin(x)$ as an inverse of $\sin(x)$?

    $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. On this interval, sine is strictly increasing from -1 to 1.

20. What restricted domain is used to define $\arctan(x)$ as the inverse of $\tan(x)$?

    $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Tangent is strictly increasing on this interval without vertical asymptotes.

21. Identify the restricted interval where $\sin(x)$ is strictly increasing and invertible.

    $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Sine increases monotonically from -1 to 1 on this interval.

22. What is the standard restricted domain for defining $\arctan(x)$ as an inverse of $\tan(x)$?

    $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. On this interval, tangent is strictly increasing through all real values.

23. What property must a trigonometric function have on a restricted domain to have an inverse?

    It must be one-to-one (strictly increasing or strictly decreasing). This ensures each output maps to exactly one input, making the function invertible.

24. What is the domain of $\arccos(x)$?

    $\left[-1,1\right]$. Since cosine's range is $[-1,1]$, this becomes arccosine's domain.

25. Evaluate $\arccos\left(-\frac{1}{2}\right)$ using the principal range of $\arccos$.

    $\frac{2\pi}{3}$. Since $\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$ and $\frac{2\pi}{3} \in [0, \pi]$.

26. Which restricted interval makes $\sin(x)$ one-to-one: $\left[0,2\pi\right]$ or $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$?

    $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Only the second interval has sine strictly monotonic throughout.

27. What restricted domain is used to define $\arcsin(x)$ as the inverse of $\sin(x)$?

    $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. On this interval, sine increases from -1 to 1, passing the horizontal line test.

28. Evaluate $\arcsin\left(\frac{1}{2}\right)$ using the principal range of $\arcsin$.

    $\frac{\pi}{6}$. Since $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ and $\frac{\pi}{6} \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.

29. What is the range of $\arcsin(x)$?

    $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. This matches the restricted domain of sine used to define its inverse.

30. What is the simplified value of $\arcsin\left(\sin\left(\frac{3\pi}{4}\right)\right)$?

    $\frac{\pi}{4}$. $\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}$, and $\arcsin(\frac{\sqrt{2}}{2})=\frac{\pi}{4}$ in range.

31. Identify the principal-value identity for $\tan(\arctan(x))$.

    $\tan(\arctan(x))=x$ for $x\in\left(-\infty,\infty\right)$. Applying tangent to its inverse returns the original value.

32. What identity expresses the inverse relationship on the domain of $\arcsin$?

    $\sin(\arcsin(x))=x$ for $x\in\left[-1,1\right]$. Composing a function with its inverse yields the identity function.

33. What property must a trigonometric function have on a restricted domain to have an inverse there?

    It must be one-to-one (strictly increasing or strictly decreasing). This ensures each input maps to exactly one output, allowing inverse construction.

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